Copied to
clipboard

## G = (C2×C8).D6order 192 = 26·3

### 173rd non-split extension by C2×C8 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — (C2×C8).D6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4×Dic3 — C12⋊Q8 — (C2×C8).D6
 Lower central C3 — C6 — C2×C12 — (C2×C8).D6
 Upper central C1 — C22 — C2×C4 — Q8⋊C4

Generators and relations for (C2×C8).D6
G = < a,b,c,d | a2=b8=1, c6=d2=a, ab=ba, ac=ca, ad=da, cbc-1=ab3, dbd-1=ab-1, dcd-1=ac5 >

Subgroups: 328 in 100 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×D4, C2×Q8, C2×Q8, C3⋊C8, C24, Dic6, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C8⋊C4, D4⋊C4, Q8⋊C4, Q8⋊C4, C4.4D4, C4⋊Q8, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, C2×Dic6, C2×D12, C6×Q8, C42.28C22, C6.D8, C24⋊C4, C2.D24, Q82Dic3, C3×Q8⋊C4, C12⋊Q8, C12.23D4, (C2×C8).D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C22×S3, C4.4D4, C8⋊C22, C8.C22, C4○D12, S3×D4, D42S3, C42.28C22, C23.11D6, Q83D6, Q16⋊S3, (C2×C8).D6

Character table of (C2×C8).D6

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 4F 4G 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 24A 24B 24C 24D size 1 1 1 1 24 2 2 2 8 8 12 12 24 2 2 2 4 4 12 12 4 4 8 8 8 8 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 1 1 1 1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 1 -1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 -1 1 1 1 1 1 -1 -1 -1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 linear of order 2 ρ5 1 1 1 1 1 1 1 1 -1 1 1 1 -1 1 1 1 -1 -1 -1 -1 1 1 1 -1 1 -1 -1 -1 -1 -1 linear of order 2 ρ6 1 1 1 1 -1 1 1 1 -1 -1 1 1 -1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ7 1 1 1 1 -1 1 1 1 -1 1 -1 -1 1 1 1 1 -1 -1 1 1 1 1 1 -1 1 -1 -1 -1 -1 -1 linear of order 2 ρ8 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ9 2 2 2 2 0 -1 2 2 2 2 0 0 0 -1 -1 -1 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 2 2 0 -1 2 2 2 -2 0 0 0 -1 -1 -1 -2 -2 0 0 -1 -1 1 -1 1 -1 1 1 1 1 orthogonal lifted from D6 ρ11 2 2 2 2 0 2 -2 -2 0 0 -2 2 0 2 2 2 0 0 0 0 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 0 -1 2 2 -2 -2 0 0 0 -1 -1 -1 2 2 0 0 -1 -1 1 1 1 1 -1 -1 -1 -1 orthogonal lifted from D6 ρ13 2 2 2 2 0 2 -2 -2 0 0 2 -2 0 2 2 2 0 0 0 0 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 2 0 -1 2 2 -2 2 0 0 0 -1 -1 -1 -2 -2 0 0 -1 -1 -1 1 -1 1 1 1 1 1 orthogonal lifted from D6 ρ15 2 -2 -2 2 0 2 2 -2 0 0 0 0 0 2 -2 -2 0 0 2i -2i -2 2 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ16 2 -2 -2 2 0 2 -2 2 0 0 0 0 0 2 -2 -2 2i -2i 0 0 2 -2 0 0 0 0 -2i 2i 2i -2i complex lifted from C4○D4 ρ17 2 -2 -2 2 0 2 2 -2 0 0 0 0 0 2 -2 -2 0 0 -2i 2i -2 2 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ18 2 -2 -2 2 0 2 -2 2 0 0 0 0 0 2 -2 -2 -2i 2i 0 0 2 -2 0 0 0 0 2i -2i -2i 2i complex lifted from C4○D4 ρ19 2 -2 -2 2 0 -1 -2 2 0 0 0 0 0 -1 1 1 2i -2i 0 0 -1 1 -√-3 -√3 √-3 √3 i -i -i i complex lifted from C4○D12 ρ20 2 -2 -2 2 0 -1 -2 2 0 0 0 0 0 -1 1 1 -2i 2i 0 0 -1 1 √-3 -√3 -√-3 √3 -i i i -i complex lifted from C4○D12 ρ21 2 -2 -2 2 0 -1 -2 2 0 0 0 0 0 -1 1 1 2i -2i 0 0 -1 1 √-3 √3 -√-3 -√3 i -i -i i complex lifted from C4○D12 ρ22 2 -2 -2 2 0 -1 -2 2 0 0 0 0 0 -1 1 1 -2i 2i 0 0 -1 1 -√-3 √3 √-3 -√3 -i i i -i complex lifted from C4○D12 ρ23 4 -4 4 -4 0 -2 0 0 0 0 0 0 0 2 -2 2 0 0 0 0 0 0 0 0 0 0 √6 √6 -√6 -√6 orthogonal lifted from Q8⋊3D6 ρ24 4 4 4 4 0 -2 -4 -4 0 0 0 0 0 -2 -2 -2 0 0 0 0 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ25 4 -4 4 -4 0 4 0 0 0 0 0 0 0 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ26 4 -4 4 -4 0 -2 0 0 0 0 0 0 0 2 -2 2 0 0 0 0 0 0 0 0 0 0 -√6 -√6 √6 √6 orthogonal lifted from Q8⋊3D6 ρ27 4 4 -4 -4 0 4 0 0 0 0 0 0 0 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ28 4 -4 -4 4 0 -2 4 -4 0 0 0 0 0 -2 2 2 0 0 0 0 2 -2 0 0 0 0 0 0 0 0 symplectic lifted from D4⋊2S3, Schur index 2 ρ29 4 4 -4 -4 0 -2 0 0 0 0 0 0 0 2 2 -2 0 0 0 0 0 0 0 0 0 0 -√-6 √-6 -√-6 √-6 complex lifted from Q16⋊S3 ρ30 4 4 -4 -4 0 -2 0 0 0 0 0 0 0 2 2 -2 0 0 0 0 0 0 0 0 0 0 √-6 -√-6 √-6 -√-6 complex lifted from Q16⋊S3

Smallest permutation representation of (C2×C8).D6
On 96 points
Generators in S96
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)
(1 94 17 64 28 59 81 48)(2 71 82 89 29 43 18 54)(3 96 19 66 30 49 83 38)(4 61 84 91 31 45 20 56)(5 86 21 68 32 51 73 40)(6 63 74 93 33 47 22 58)(7 88 23 70 34 53 75 42)(8 65 76 95 35 37 24 60)(9 90 13 72 36 55 77 44)(10 67 78 85 25 39 14 50)(11 92 15 62 26 57 79 46)(12 69 80 87 27 41 16 52)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 38 7 44)(2 37 8 43)(3 48 9 42)(4 47 10 41)(5 46 11 40)(6 45 12 39)(13 53 19 59)(14 52 20 58)(15 51 21 57)(16 50 22 56)(17 49 23 55)(18 60 24 54)(25 69 31 63)(26 68 32 62)(27 67 33 61)(28 66 34 72)(29 65 35 71)(30 64 36 70)(73 92 79 86)(74 91 80 85)(75 90 81 96)(76 89 82 95)(77 88 83 94)(78 87 84 93)

G:=sub<Sym(96)| (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96), (1,94,17,64,28,59,81,48)(2,71,82,89,29,43,18,54)(3,96,19,66,30,49,83,38)(4,61,84,91,31,45,20,56)(5,86,21,68,32,51,73,40)(6,63,74,93,33,47,22,58)(7,88,23,70,34,53,75,42)(8,65,76,95,35,37,24,60)(9,90,13,72,36,55,77,44)(10,67,78,85,25,39,14,50)(11,92,15,62,26,57,79,46)(12,69,80,87,27,41,16,52), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,38,7,44)(2,37,8,43)(3,48,9,42)(4,47,10,41)(5,46,11,40)(6,45,12,39)(13,53,19,59)(14,52,20,58)(15,51,21,57)(16,50,22,56)(17,49,23,55)(18,60,24,54)(25,69,31,63)(26,68,32,62)(27,67,33,61)(28,66,34,72)(29,65,35,71)(30,64,36,70)(73,92,79,86)(74,91,80,85)(75,90,81,96)(76,89,82,95)(77,88,83,94)(78,87,84,93)>;

G:=Group( (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96), (1,94,17,64,28,59,81,48)(2,71,82,89,29,43,18,54)(3,96,19,66,30,49,83,38)(4,61,84,91,31,45,20,56)(5,86,21,68,32,51,73,40)(6,63,74,93,33,47,22,58)(7,88,23,70,34,53,75,42)(8,65,76,95,35,37,24,60)(9,90,13,72,36,55,77,44)(10,67,78,85,25,39,14,50)(11,92,15,62,26,57,79,46)(12,69,80,87,27,41,16,52), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,38,7,44)(2,37,8,43)(3,48,9,42)(4,47,10,41)(5,46,11,40)(6,45,12,39)(13,53,19,59)(14,52,20,58)(15,51,21,57)(16,50,22,56)(17,49,23,55)(18,60,24,54)(25,69,31,63)(26,68,32,62)(27,67,33,61)(28,66,34,72)(29,65,35,71)(30,64,36,70)(73,92,79,86)(74,91,80,85)(75,90,81,96)(76,89,82,95)(77,88,83,94)(78,87,84,93) );

G=PermutationGroup([[(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48),(49,55),(50,56),(51,57),(52,58),(53,59),(54,60),(61,67),(62,68),(63,69),(64,70),(65,71),(66,72),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84),(85,91),(86,92),(87,93),(88,94),(89,95),(90,96)], [(1,94,17,64,28,59,81,48),(2,71,82,89,29,43,18,54),(3,96,19,66,30,49,83,38),(4,61,84,91,31,45,20,56),(5,86,21,68,32,51,73,40),(6,63,74,93,33,47,22,58),(7,88,23,70,34,53,75,42),(8,65,76,95,35,37,24,60),(9,90,13,72,36,55,77,44),(10,67,78,85,25,39,14,50),(11,92,15,62,26,57,79,46),(12,69,80,87,27,41,16,52)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,38,7,44),(2,37,8,43),(3,48,9,42),(4,47,10,41),(5,46,11,40),(6,45,12,39),(13,53,19,59),(14,52,20,58),(15,51,21,57),(16,50,22,56),(17,49,23,55),(18,60,24,54),(25,69,31,63),(26,68,32,62),(27,67,33,61),(28,66,34,72),(29,65,35,71),(30,64,36,70),(73,92,79,86),(74,91,80,85),(75,90,81,96),(76,89,82,95),(77,88,83,94),(78,87,84,93)]])

Matrix representation of (C2×C8).D6 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 46 0 0 0 0 0 0 46 0 0 0 0 0 0 34 68 39 5 0 0 5 39 68 34 0 0 34 68 34 68 0 0 5 39 5 39
,
 27 8 0 0 0 0 0 46 0 0 0 0 0 0 66 59 14 7 0 0 14 7 66 7 0 0 14 7 7 14 0 0 66 7 59 66
,
 46 0 0 0 0 0 18 27 0 0 0 0 0 0 34 39 39 34 0 0 5 39 68 34 0 0 39 34 39 34 0 0 68 34 68 34

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,34,5,34,5,0,0,68,39,68,39,0,0,39,68,34,5,0,0,5,34,68,39],[27,0,0,0,0,0,8,46,0,0,0,0,0,0,66,14,14,66,0,0,59,7,7,7,0,0,14,66,7,59,0,0,7,7,14,66],[46,18,0,0,0,0,0,27,0,0,0,0,0,0,34,5,39,68,0,0,39,39,34,34,0,0,39,68,39,68,0,0,34,34,34,34] >;

(C2×C8).D6 in GAP, Magma, Sage, TeX

(C_2\times C_8).D_6
% in TeX

G:=Group("(C2xC8).D6");
// GroupNames label

G:=SmallGroup(192,353);
// by ID

G=gap.SmallGroup(192,353);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,120,1094,135,184,570,297,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=1,c^6=d^2=a,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^3,d*b*d^-1=a*b^-1,d*c*d^-1=a*c^5>;
// generators/relations

Export

׿
×
𝔽